Zero Temperature Limits of Gibbs-Equilibrium States for Countable Alphabet Subshifts of Finite Type

Abstract

Let A be a finitely primitive subshift of finite type on a countable alphabet. For appropriate functions f : A → IR, the family of Gibbs-equilibrium states (µ tf ) t 1 for the functions tf is shown to be tight. Any weak * -accumulation point as t → ∞ is shown to be a maximizing measure for f .

Ergodic Optimization for Renewal Type Shifts

“…This is not the case when considering countable Markov shifts. Jenkinson, Mauldin and Urbański [3,4] have proved that for a certain class of countable Markov shifts, where the thermodynamic formalism is similar to the one observed in finite state Markov shifts, there is always an optimal measure. The class R has no intersection with the class considered by them.…”

“…For a regular potential φ defined on a Markov shift belonging to the class R, it is possible to show using results from [6] that there exists a critical value t c ∈ (0, ∞] such that the pressure function P (tφ) is real analytic on (0, t c ) and linear on (t c , ∞). In this note we tie together results from Jenkinson-Mauldin-Urbański [3] and Sarig [6] to prove the following Theorem 1.1. Let (Σ, σ) ∈ R and φ : Σ → R a locally Hölder potential such that sup φ < ∞, then (1) If t c = ∞ then there exist φ−optimal measures.…”

Sturmian maximizing measures for the piecewise-linear cosine family

“…[29] for an overview). Most of this work has concerned theoretical aspects of the subject, including abstract information on the nature of maximizing measures [8,10,11,13,19,30,36,37,39,40,41,42,44,47,48,49,51], approximation of maximizing measures [9,15,18], connections with thermodynamic formalism [12,17,27,28,33,35,38,46], and connections with partial orders on M [2,31,32,34].…”

The Mañé–Conze–Guivarc’h lemma for intermittent maps of the circle

“…The study of maximizing measures has recently become the focus of significant research interest. While early articles of Bousch and Jenkinson [2,14] were motivated by abstract questions concerning the geometric structure of the set of measures M T , questions relating to maximizing measures have also appeared in research into chaotic control [13,25], Livšic-type theorems [6], thermodynamic formalism [9,15,16], Tetris heaps [7], and the Lagarias-Wang finiteness conjecture in linear algebra [7].…”